Critical surfaces for general inhomogeneous bond percolation problems

We present a method of general applicability for finding exact values or accurate approximations of bond percolation thresholds for a wide class of lattices. To every lattice we systematically associate a polynomial, the root of which in [0, 1] is the conjectured critical point. The method makes the correct prediction for every exactly solved problem, and comparison with numerical results shows that it is very close, but not exact, for many others. We focus primarily on the Archimedean lattices, in which all vertices are equivalent, but this restriction is not crucial. Some results we find are kagome: p c = 0.524 430..., (3, 12 2 ): p c = 0.740 423..., (3 3 , 4 2 ): p c = 0.419 615..., (3, 4, 6, 4): p c = 0.524 821..., (4, 8 2 ): p c = 0.676 835..., (3 2 , 4, 3, 4): p c = 0.414 120... . The results are generally within 10 − 5 of numerical estimates. For the inhomogeneous checkerboard and bow-tie lattices, errors in the formulae (if they are not exact) are less than 10 − 6 .